17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (sin² 3x) / x²

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Calculate the value of the given limit. Apply Lajapatal's rule if necessary. The limit as x approaches 0 of x2 all divided by sin square of 5 X. Fantastic. So it appears for this particular prompt, we're asked to calculate the value of this given limit, and that is our final answer that we're ultimately trying to solve for. So now that we know that we're ultimately trying to evaluate this particular limit, let us read off our multiple choice answers to see what our final answer might be. A is 1 divided by 25, B is 1 divided by 5, C is 5, and D is 25. OK. So first off, we're given a hint that we can apply Lahopital's rule if necessary, and since we're given this hint, let us use the direct substitution method first in order to see if this particular limit when we use direct substitution, will give us an indeterminate form as a final answer. So with that said, as we should recall a note, an indeterminate forms will be 0 divided by 0 or Infinity divided by infinity. So when we use direct substitution, we need to look at our limit that's given to us. In this case we're told the limit as x approaches 0, so we can use direct substitution to state that X equals 0 for this particular case. So let's use direct substitution to see if we receive an indeterminate form. So we need to take 0 squad, because we need to plug in a value of 0 in for X, divided by sin squad of 5 multiplied by 0. Fantastic. And when we do that, we will indeed get an indeterminate form means we will get 0 divided by 0 as an answer. And because we get this indeterminate form, we need to use Lahoital's rule to solve for this limit because the expression leads to an indeterminate form when we use the direct substitution method. So, for evaluating limits with indeterminate forms such as 0 divided by 0 or infinity divided by infinity, we need to recall and apply Lahapatal's rule. As we should recall and note, Lahabital's rule states that for calculating limits with indeterminate forms, we need to differentiate the numerator and denominator separately, and we need to repeat this process until we stop getting an indeterminate form as an answer. So, the numerator once again is the top expression of our division bar, and the denominator is the bottom expression of our division bar. And what does it mean when they say repeat the process until we stop getting an indeterminate form. So essentially, we need to keep taking the first derivative of the numerator and denominator as many times as it takes. To perform the direct substitution method, so when we take the first derivative and we apply the direct substitution method, and if it's still an indeterminate form as a final answer, we need to keep doing the derivatives of the numerator and denominator until we get an answer that's not 0 divided by 0 or infinity divided by infinity. So, with that said, we need to recall and apply Lahapato's rule to solve for this particular limit. So with that said, we can go ahead and write that the limit as X approaches 0 of X2 divided by sine squad of 5 X. fantastic. Which will be equal to the limit as X approaches 0 of. X divided by sin of 5 X all to the power of 2, let me simplify, which we can say that this is equal to the limit as X approaches 0 of X divided by sine of 5 X. All to the power of 2, so even the limit approaching 0 is inside of these parentheses to the power of 2. So now our first step is we need to find the derivatives of the numerator and the denominator, and we need to differentiate the numerator and denominator with respect to X. So let's start with the numerator. So we take D by DX, which D by DX is a fancy way of saying we're taking the derivative with respect to X. And D by DX of X is going to be equal to 1. And that is for our numerator and for the denominator, we will get D by DX of sign. Of 5 X, and when we perform that derivative, we will find that it's equal to cosine of 5 X multiplied by D by DX of 5 X, which will be All equal to drum roll. 5 multiplied by cosine of 5 X. Fantastic. So, moving right along here, now we can recall and apply Lahapital's rules, so we could say that the limit as X approaches 0 of X2 divided by sin squad of 5 X. is equal to the limit as X approaches 0 of. X divided by Sign of. 5 X, so the sign of 5 X. All to the power of 2 is equal to parenthesis, the limit as X approaches 0. Of X divided by sin of 5 X all to the power of 2, which will be all equal to parenthesis, the limit as X approaches 0 of, so X approaches 0 of 1 divided by 5 multiplied by cosine of 5 X Cal to the power 2, which is going to be equal to. 1 divided by 5 multiplied by cosine of 5 multiplied by 0, because now we need to use a direct substitution method to plug in a value of 0 in our value of X, and don't forget that this is all. The power of 2. So when we plug that into our calculator, when we plug in 1 divided by 5 multiplied by cosine of 5 multiplied by 0, we will receive an answer of 1 divided by 25 when we write it in fraction form. And that will be our final answer. Hooray, we did it. And this corresponds with multiple choice answer letter A, which once again states that our final answer is 1 divided by 25. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (sin² 3x) / x²

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Limits

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